Theorem pm5.32dar

Description: Reverse distribution of implication over biconditional (deduction form). (Contributed by Zhi Wang, 6-Sep-2024)

	Ref	Expression
Hypothesis	pm5.32dar.1	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\leftrightarrow \left(\psi \wedge \theta \right)\right)$$
Assertion	pm5.32dar	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm5.32dar.1	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\leftrightarrow \left(\psi \wedge \theta \right)\right)$$
2		pm5.32	$$⊢\left(\psi \to \left(\chi \leftrightarrow \theta \right)\right)\leftrightarrow \left(\left(\psi \wedge \chi \right)\leftrightarrow \left(\psi \wedge \theta \right)\right)$$
3	1 2	sylibr	$$⊢\phi \to \left(\psi \to \left(\chi \leftrightarrow \theta \right)\right)$$
4	3	imp	$$⊢\left(\phi \wedge \psi \right)\to \left(\chi \leftrightarrow \theta \right)$$